Question

two parallel lines are cut by a transversal as shown below. suppose ( m angle 8 = 81 ^ { circ } ). find ( m angle 1 ) and ( m angle 3 ).

Explanation:

Step1: Find \(m\angle 1\)

\(\angle 1\) and \(\angle 8\) are alternate exterior angles.
Since alternate exterior angles are congruent when two parallel lines are cut by a transversal, \(m\angle 1=m\angle 8 = 81^{\circ}\).

Step2: Find \(m\angle 3\)

\(\angle 3\) and \(\angle 8\) are supplementary angles (because \(\angle 3\) and \(\angle 4\) are a linear - pair, \(\angle 4=\angle 8\) as alternate interior angles).
We know that if two angles are supplementary, \(m\angle 3 + m\angle 8=180^{\circ}\).
Substitute \(m\angle 8 = 81^{\circ}\) into the equation: \(m\angle 3=180^{\circ}-m\angle 8\).
\(m\angle 3 = 180^{\circ}-81^{\circ}=99^{\circ}\).

Answer:

\(m\angle 1 = 81^{\circ}\)
\(m\angle 3 = 99^{\circ}\)